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\title{作业二:论文排版 }


\author{金郁香 \\信息与计算科学 3220103054}

\begin{document}

\maketitle


\section{问题描述}

对要求章节进行翻译与排版

\section{ 不可压缩Navier-Stokes方程}
不可压缩流体的二维流场完全由速度矢量$ q = (u(x, y), v(x, y)) \in R^2 $和压力$ e , p(x, y) \in R $来描述。\cite{reference1}这些函数是以下守恒定律的解（例如，见Hirsch，1988）：
\begin{itemize}
\item  质量守恒：
  \begin{equation}\label{eq1}
    div(q)=0 \tag{12.1}
  \end{equation}
  或者使用diversince1算子的显式形式编写，
  \begin{equation}\label{eq2}
    \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0\tag{12.2}
  \end{equation}
\item  紧致形式的动量守恒方程
  \begin{equation}\label{eq3}
    \frac{\partial q}{\partial t} + \nabla \cdot (q \otimes q) = -\mathcal{G}p + \frac{1}{Re}\Delta q \tag{12.3} \end{equation}
  或者以明确的形式，
  \begin{equation}\label{eq4}
  \left\{
    \begin{array}{lr}
     \frac{\partial u}{\partial t} + \frac{\partial u^2}{\partial x} + \frac{\partial uv}{\partial y} = -\frac{\partial p}{\partial x} + \frac{1}{Re} (\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} ),&\\
     \frac{\partial v}{\partial t} + \frac{\partial uv}{\partial x} + \frac{\partial v^2}{\partial y} = -\frac{\partial p}{\partial y} + \frac{1}{Re} ( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2}),&
    \end{array}
    \right.
    \tag{12.4}
  \end{equation}
  前面的方程以无量纲形式编写，使用以下缩放变量：
  \begin{equation}
    x = \frac{x^*}{L}, \quad y = \frac{y^*}{L}, \quad u = \frac{u^*}{V_0}, \quad v = \frac{v^*}{V_0}, \quad t = \frac{t^*}{L/V_0}, \quad p = \frac{p^*}{\rho_0 V_0^2}\tag{12.5}
  \label{pythagorean}
  \end{equation}
其中上标（*）表示以物理单位测量的变量。这个常数L、$V_0$分别是表征模拟流动的参考长度和速度。无量纲数Re称为雷诺数数字并量化惯性（或对流）项的相对重要性以及流动中的粘性$（$或扩散$）^3$项：
  \begin{equation} Re = \frac{V_0L}{\nu} \tag{12.6}
  \label{pythagorean}
  \end{equation}
其中，$\mathcal v$是流动的运动粘度。
  总之，偏微分方程的Navier-Stokes系统将在数值上本项目中解决的问题由\ref{eq2}和\ref{eq4}定义；初始条件（在t=0），并且边界条件将在下面的部分中讨论。
  
\end{itemize}

\section{计算域、交错网格和边界条件}
数值求解Navier-Stokes方程相当简单，通过考虑具有周期性的矩形域$L_x×L_y$边界条件无处不在，速度q(x,y)和压力场p(x,y)在数学上表示为 
\begin{align*}
q(0, y) &= q(L_x, y), \quad p(0, y) = p(L_x, y), \quad \forall y \in [0, L_y] \tag{12.7} \label{pythagorean} \\
q(x, 0) &= q(x, L_y), \quad p(x, 0) = p(x, L_y), \quad \forall x \in [0, L_x] \tag{12.8} \label{pythagorean}
\end{align*}\

计算解的点分布在域遵循矩形和均匀的2D网格。因为不是所有的变量在我们的方法中，共享相同的网格，我们首先定义了一个主网格（请参见流体动力学：求解二维Navier-Stokes方程,如图\ref{figleft}）通过分别沿着x取$n_x$个计算点生成，沿着y的$n_y$个点：
\begin{align*}
  x_c(i) &= (i - 1)\delta x, \quad \delta x = \frac{{L_x}}{{n_x}}, \quad i = 1, \ldots, n_x  \tag{12.9} \label{pythagorean} \\\
y_c(j) &= (j - 1)\delta y, \quad \delta y = \frac{{L_y}}{{n_y}}, \quad j = 1, \ldots, n_y\tag{12.10} \label{pythagorean}
\end{align*}

\begin{center}
  \begin{minipage}{0.3\textwidth}
    \begin{tikzpicture}
      \draw[very thick] (-1, 0) -- (6.5, 0); % 绘制 x 轴
  \draw[very thick] (-1, 0) -- (-1, 5.5); % 绘制 y 轴
  \draw[very thick, <->] (-1, 2.5) -- (0.5,2.5);
  \draw[very thick, <->] (3, 2.5) -- (4,2.5);
  \draw[very thick, <->] (1.5, 0) -- (1.5, 1.5);
  \draw[very thick, <->] (1.5, 3.5) -- (1.5, 5);
  \draw[very thick,red] (-1, 0)--(-1, 5)--(4,5)--(4,0)--(-1,0);
  \node[below] at (-1,0) {${0}$};
  \node[left] at (-1,0) {${0}$};
  \node[below] at (2,0) {${x}$};
  \node[left,rotate=90] at (-1.25,2.5) {${y}$};
  \node[above] at (-1,6) {${L_x}$};
  \node[above left] at (4,0) {${L_y}$};
  \node[above]  at (0.5,2.5) {periodicity};  
  \node[above] at (4.5,2.5) {periodicity};
  \node[right] at (1.5, 0.6) {periodicity};
  \node[right] at (1.5, 4) {periodicity};
  
    \end{tikzpicture}
  \end{minipage}
  \hspace{2cm}
  \begin{minipage}{0.3\textwidth}
    \begin{tikzpicture}
        \draw[very thick,black,-] (0, 0) -- (7, 0);
  \draw[very thick,black,-] (0, 0) -- (0, 7);
  \draw[very thick,green,-] (0, 6) -- (6, 6);
  \draw[very thick,green,-] (6, 0) -- (6, 6);
  \draw[very thick,green,-] (0, 2) -- (6, 2);
  \draw[very thick,green,-] (2, 0) -- (2, 6);
  \draw[very thick,green,-] (4, 0) -- (4, 6);
  \draw[very thick,green,-] (0, 4) -- (6, 4);
  \draw[thick,black,dotted] (0, 3) -- (6, 3);
  \draw[thick,black,dotted] (3, 0) -- (3, 6);

  \node[below] at (7, 0) {${X}$}; % 在指定位置添加标记字母
  \node[left, rotate=90] at (-0.5, 7) {${Y}$}; % 在指定位置添加标记字母
  \node[left] at (0, 0) {${0}$};
  \node[below] at (0, 0) {${0}$};
  \node[above] at (6.5, 0) {${L_x}$};
  \node[right] at (0, 6.5) {${L_y}$};
  \draw[fill=black] (3, 2) circle (2pt);
  \draw[fill=black] (3, 3) circle (2pt); % 绘制填充黑色的点
  \draw[fill=black] (2, 3) circle (2pt); % 绘制填充黑色的点
  \node[above right] at (3, 3) {p(i,j)};
  \node[above left] at (2, 3) {u(i,j)};
  \node[below right] at (3, 2) {v(i,j)};
  \node[left] at (0, 3) {$y_m(j)$};
  \node[left] at (0, 2) {$y_c(j)$};
  \node[left] at (0, 4) {$y_c(j+1)$};
  \node[below,rotate=90] at (1.8, -0.7) {$x_c(i)$};
  \node[below,rotate=90] at (2.8, -0.7) {$x_m(i)$};
  \node[below,rotate=90] at (3.8, -0.7) {$x_c(i+1)$}; 
    \end{tikzpicture}
  \end{minipage}
\end{center}
\label{figleft}
(图3,计算域、交错网格和边界条件)

二次网格由一次网格单元的中心定义：
\begin{align*}
x_m(i) &= \left(i - \frac{1}{2}\right)\delta x, \quad i = 1, \ldots, n_{xm}, \tag{12.11} \label{pythagorean} \\\
y_m(j) &= \left(j - \frac{1}{2}\right)\delta y, \quad j = 1, \ldots, n_{ym}, \tag{12.12} \label{pythagorean}
\end{align*}

其中我们使用了缩写符号$n_{xm} = n_x - 1, n_{ym} = n_y - 1$.在既定范围内定义为矩形$[ [x_c(i), x_c(i + 1)] \times [y_c(j), y_c(j + 1)] ]$，未知变量u、v、p将被计算为不同空间位置的解决方案：

• $u(i, j) \approx u\left(x_c(i), y_{m}(j)\right)$(单元格的西边界）

• $v(i, j) \approx v\left(x_{m}(i), y_c(j)\right)$（单元格的南边界）

• $p(i, j) \approx p\left(x_{m}(i), y_{m}(j)\right)$（单元格的中心）

这种变量交错排列的优点是压力和速度之间的耦合。它也有利于（请参阅本章结束）去避免遇到一些稳定性和收敛性问题具有并置排列（其中计算所有变量在相同的网格点处）。

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